In the United States, where I live, coins don’t make much sense. Our nickels are worth less than dimes, but are larger and thicker. There’s more value in the raw material of some pennies than the cent they’re worth. It is all a little off-kilter.
Assuming you wanted the value of coins to be commensurate with their volume, a coin with twice of another should be worth the radius should be worth four times the value. That’s because every time you double a dimension, you double the overall size, and doubling the radius effectively doubles both the length and height. If the coin was also twice as wide, it would use eight times as much metal.
There are lots of fun puzzles that would come up with a set of coins that were in this kind of 2 to 1 proportion. So let’s assume we had coins like these. I’ll call the small ones “alphas” and the big ones “betas.”
The puzzles I want to share today are some of my favorites, because they are so counterintuitive.
Puzzle 1
Take two A coins, set one atop the other, and roll the top coin around the bottom one until it arrives back where it started. How many revolutions will it make on this journey?
Puzzle 2
Now put the A coin on top of the B coin, and roll it until it arrives back where it started. How many revolutions will it make?
Puzzle 3
Now consider what would happen if we asked the same questions about the B coin rolling around the A coin. How many revolutions will the B coin make on its journey?
The possibilities for extensions of this question are essentially limitless. Here’s one that jumps to mind:
Research question: you have two coins with radii in an n:m ratio. Can you determine a formula for the number of revolutions one will turn as it rolls along the entire perimeter of the other?
Solutions
One approach to solving these puzzles is to actually cut the circles out of paper or cardboard and just do it very slowly.
The surprise jumps out right away: the coin that rolls along the other revolves twice, not just once. One way you can see this in the diagram itself is to consider the point of contact between the coins. The arrow pointing right will touch the right side of the stationary coin as it rolls. That means that in a quarter of its journey, the coin rotates 180 degrees.
Solution to puzzle 2.
We can solve all the puzzles the way we did the first. However, I’d like to find a more powerful perspective.
Here’s another approach: imagine that the rotating coin is rotating at a constant rate, given by its speed. We just need to find out how far it goes. What confuses this approach is that different points on the coin go different distances, and travel at different speeds. But what if we track the center of the circle?
In this case, the path the centre of coin alpha travels is three times its diameter. That means its own diameter will uncoil three times as it rotates around beta, which means it will make three revolutions!
Solution to puzzle 3.
And in fact, this method seems to keep working. For the larger coin, the distance the center of B travels is in a 3:2 ratio with the diameter of B, which means B turns 1.5 revolutions as it travels around A.
My own experience with this argument is that I have a hard time trusting it. I convince myself it works, then I doubt my own argument. Do you find it convincing? Can you use it to solve the general case of coins of different sizes rotating around each other? Do you believe your argument?
Happy puzzling!
